Editing Local boundedness (section)

===Locally bounded topological vector spaces===
{{Main|Seminormed space}}

A [[subset]] <math>B \subseteq X</math> of a topological vector space (TVS) <math>X</math> is called '''[[Bounded set (topological vector space)|bounded]]''' if for each neighborhood <math>U</math> of the origin in <math>X</math> there exists a real number <math>s > 0</math> such that
<math display=block>B \subseteq t U \quad \text{ for all } t > s.</math>
A '''{{visible anchor|locally bounded TVS}}''' is a TVS that possesses a bounded neighborhood of the origin. 
By [[Kolmogorov's normability criterion]], this is true of a locally convex space if and only if the topology of the TVS is induced by some [[seminorm]]. 
In particular, every locally bounded TVS is [[Metrizable topological vector space|pseudometrizable]].